Chapter 2 Math 3201 1 Chapter 2: Counting Methods: • Solving problems that involve the Fundamental Counting Principle • Understanding and simplifying expressions involving factorial notation • Solving problems that involve permutations • Solving problems that involve combinations Section 2.1: Counting Principles: Example 1: A toy manufacturer makes a wooden toy in three parts: Part 1: the top part may be colored red, white or blue Part 2: the middle part may be orange or black, Part 3: the bottom part may be yellow, green, pink, or purple Use a tree diagram to determine how many different colored toys can be produced:
Transcript
Chapter 2 Math 3201 1
Chapter 2: Counting Methods: • Solving problems that involve the Fundamental Counting Principle • Understanding and simplifying expressions involving factorial notation • Solving problems that involve permutations • Solving problems that involve combinations
Section 2.1: Counting Principles: Example 1: A toy manufacturer makes a wooden toy in three parts:
Part 1: the top part may be colored red, white or blue
Part 2: the middle part may be orange or black,
Part 3: the bottom part may be yellow, green, pink, or purple
Use a tree diagram to determine how many different colored toys can be produced:
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Make a conjecture about how you can use multiplication only to arrive at the number of
different colored toys possible.
. . =
top middle bottom colored toys
When the sample space is very large, the task of listing and counting all the outcomes in a given situation is unrealistic.
The Fundamental Counting Principle helps us find the number of outcomes without counting each one. The Fundamental Counting Principle
• Applies when tasks are related by AND.
• If one task can be performed in a ways and another task can be performed in b ways, then both tasks can be completed in 𝑎 ∙ 𝑏 ways
Example 2: Determine the number of distinguishable four letter arrangements that can be formed from the word ENGLISH if:
a. letters can be repeated?
b. no letters are repeated and:
i) there are no further restrictions?
. . . =
ii) the first letter must be E?
1 . . . = E
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Example 3: In each case, how many 3–digit whole numbers can be formed using the digits 1, 2, 3, 4 and 5?
a) Repetitions are allowed
b) Repetitions are not allowed.
Example 4: How many ways can 5 people stand in a line for a photo?
Example 5: A standard deck of cards contains 52 cards.
Count the number of possibilities of drawing a single card and getting:
a. either a black face card or an ace
(What does ‘or’ mean?, can you use the fundamental counting principle?)
Can’t use the fundamental counting principle because a black face card
and an ace is not possible on a single draw.
b. either a red card or a 10
Include the elements of the union of both sets and exclude the elements in the intersection. Needed to subtract the overlap (red 10’s) because we do not want to count
them twice.
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In Summary:
• The Fundamental Counting principle applies when tasks are related by the word
AND.
• The Fundamental Counting principle does not apply when tasks are related by the
word OR.
o If the tasks are mutually exclusive, they involve two disjoint sets:
𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵)
o If the tasks are not mutually exclusive, they involve two sets that are not disjoint:
𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵)− 𝑛(𝐴 ∩ 𝐵)
Practice problems: Page 73-74 #2-7,9,11,13,15,20
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Section 2.2: Introducing Permutations and Factorial Notation: Factorial Notation Consider how many ways there are of arranging 6 different books side by side on a shelf. In this example we have to calculate the product 6x5x4x3x2x1. In mathematics this product is denoted by 6! (“factorial” or “factorial 6”) In general, n!=n(n-1)(n-2)(n-3)….(3)(2)(1), where 𝒏 ∈ 𝑵 Example 1: Use the factorial key on your calculator to solve 6!
Permutations An arrangement of a set of objects in which the order of the objects is important Example 2: How many permutations are there of the letters of the word: a. REGINA: 6! =
b. KELOWNA: 7! =
Evaluating and Simplifying Factorial Expressions: Example 3: To simplify10!
7!, there are several approaches:
a. Use you calculator: = 720 b. By Cancellation:
10! 10 9 8 7!7! 7!
× × ×= =10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 720
Extra Practice: Permutations and Factorial Notation
1) Evaluate each of the following: a) 5! b) !18!20 c)
!9!10 d)
!910!10
×
2) Simplify each of the following expressions.
a) )!1(
!−nn b)
)!2()!1(
−−
aa c)
)!2(!−aa d)
)!1()!1(
−+
aa
3) On the assembly line at factory, six digit serial numbers are assigned to products according to the following regulations: only the digits 4 to 9 are used; no digit may be used twice in the same serial number.
a) Use the Fundamental Counting Principle to calculate the number of possible serial numbers under the system.
b) Write this answer using factorial notation.
4) Solve for n, where ∈n N.
a) 9!
)!1(=
+n
n b) (𝑛+1)!(𝑛−1)!
= 42 c) (𝑛+5)!(𝑛+3)!
= 30 d) (𝑛+4)!(𝑛+2)!
= 2
5) List all of the 2-arrangements of the symbols {+, -, x}
Answers: 1a) 120 b) 380 c) 10 d) 1 2a) n b) a-1 c) aa −2 d) aa +2 3) 6 x 5 x 4 x 3 x 2 x 1 = 720 b) 6!
4 a) 8 b) 6 c) 1 d) -2 5) +-, -+, +x, x+, -x, x-
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Section 2.3: Permutations When All Objects are Distinguishable Permutations An arrangement of a set of objects in which the order of the objects is important
The number of permutations of “n” different objects taken “r” at a time is:
!( )!
nPn r n r=−
where 0 ≤ 𝑟 ≤ 𝑛
Example 1: (A) Use the n rP key on your calculator to evaluate 8 3P .
(B) Then solve using factorials.
!( )!
nPn r n r=−
8P3 =8!
(8−3)!
Example 2: evaluate the following:
(A) 6P4 (B) 16P3
(C) 8P8 (D) 7P0
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Example 3: How many three-letter permutations can you make with the letters of the word MATH?
Defining 0! If we replace r by n in the above formula we get the number of permutations of n objects taken n at a time. This we know is n!
! !!( )! 0!n n
n np nn n
= = =−
For this to be equal to n! the value of 0! Must be 1.
0! Is defined to have a value of 1
2 2 2!P =
Permutations with Restrictions
In many problems restrictions are placed on the order in which objects are arranged. In this type of situation, deal with the restrictions first.
Example 4: In a South American country, vehicle license plates consist of any 2 different letters followed by 4 different digits. Find how many different license plates are possible using: Fundamental Counting Principle:
Permutations:
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Example 5: In how many different ways can 3 girls and 4 boys be arranged in a row if no two people of the same gender can sit together?
Example 6: In how many ways can all of the letters of the word ORANGES be arranged if:
a) there are no further restrictions:
b) the first letter must be an N? :
c) the vowels must be together in the order O, A, and E:
(You need to multiply by 5 because the OAE can fill into any of the 5 slots)
Practice problems: Page 93 #1-8
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Example 7: How many arrangements of the word FAMILY exist if AL must always be together? Example 8: At a used car lot, seven different models of cars are to be parked side close to the street for viewing.
(A) The three red cars must be parked so that there is a red car at each end and the third red car must be exactly in the middle. How many ways can the seven cars be parked?
(B) The three red cars must be parked side by side. How many ways can the seven cars be parked?
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Example 9: To open the garage door Mary uses a keypad containing the digits 0 through 9. The password must be at least a 4 digit code to a maximum of 6 digits, and each digit can only be used once in the code. How many different codes are possible? Example 10: Solving a permutation equation: (A) nP2 = 42 (B) n-1P2 = 12
Practice problems: Page 94 #10,12,13,15
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2.3 Extra Practice Permutations:
Permutations when all objects are distinguishable nPr = 𝑛!(𝑛−𝑟)!
1. Determine all the possible 7-song playlists, then 8-song playlists, and finally 9-song playlists that Matt can create from 10 songs? How does the value of nPr change as r gets closer to n?
2. There are 10 movies playing at Empire Cinemas. In how many ways can you see two of them consecutively?
3. A) In how many ways can the letters of the word GRAPHITE be arranged?
B) How many three letter arrangements can be made from the word GRAPHITE?
Solving a Permutation problem with Conditions
4. Find the number of permutations of the letters in the word KITCHEN if A) the letters K, C, and N must be together but not necessarily in that order. B) the vowels must be kept together.
5. In how many ways can 3 girls and 4 boys be arranged in a row if no two people of the same
gender can sit together?
6. Six actors and eight actresses are available for a play with four male roles and three female roles. How many different cast lists are possible?
Solving a Permutation Problem involving Cases
If a counting problem has one or more conditions that must be met,
*Consider each case that each condition creates, and
*Add the number of ways each case can occur to determine the total number of outcomes.
7. Tania needs to create a password for a social networking website. • The password can use any digits 0 to 9 and/or any letters of the alphabet. • The password is case sensitive, so she can use both lower and uppercase letters. • A password must be at least 5 characters to a maximum of 7 characters. • Each character must be used only once in the password.
How many different permutations are possible?
Hint Find the total numbers of characters first: ___________________ (This represents n)
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8. How many numbers can be made from the digits 2, 3, 4, and 5 if no digit can be repeated?
(Hint: consider 4 cases – four digit numbers, three digit numbers, two digit numbers, one digit numbers.)
Comparing arrangements created with and without repetition
9. A social insurance number (SIN) in Canada consists of a nine-digit number that uses the digits 0 to 9.
A) If there are no restrictions on the digits selected for each position in the number, how many SINs can be created if each digit can be repeated?
B) How does this compare with the number of SINs that can be created if no repetition is allowed?
10. Mrs. Pi has made up 12 multiple choice questions for a math quiz, where the answers A, B, C, and D are possible. How many different answer keys are possible if there are no restrictions?
Solving Equations of the form nPr = k
11. Solve for n: A) nP2 = 30 B) n-1P2 = 12 C) nP2 = 56 D) n+1P2 = 20
Answers: 1. 604800, 1814400, 3628800 as r gets closer to n the value of nPr increases. 2. 90 3a) 40320 3b) 336 4a) 720 b)1440 5. 144 6. 120960 7. 2.52x1012 8. 64 9a) 109 b) 3628800 10. 16777216 11.a) 6 b) 5 c) 8 d) 4
Mid-Chapter Review – pg. 96-97
Assignment: pg. 97 #1-9,11,12 IN-CLASS ASSIGNMENT TO HERE
Chapter 2 SECTIONS 2.1-2.3
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Section 2.4: Permutations When Objects are Identical:
Consider the words BIKE and BOOK. How does the number of ways that the four letters in each word can be arranged differ? Why?
There are fewer permutations when some of the objects in a set are identical
Permutations with Repetitions The following formula gives the number of permutations when there are repetitions:
Example 1: Find the number of permutations of the letters of the word:
a.) VANCOUVER: there are 2V’s:
b.) MATHEMATICAL: 2M’s, 3A’s, 2T’s:
The number of permutations of n objects, where a are the same of one type, b are the same of another type, and c are the same of yet another type, can be represented by the expression below
P= !! ! !n
a b c
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Example 2: How many arrangements of the word POPPIES can be made under each of the following conditions?
a) without restrictions:
b) if each arrangement begins with a P:
P OPPIES
The first letter is a P, and the next 6 letters can be in any arrangement (remember there is a repetition of 2P’s within the remaining 6 letters)
=
c) if all the P’s are together:
PPP OIES
Example 3: Brett bought a carton containing 10 mini boxes of cereal. There are 3 boxes of Corn Flakes, 2 boxes of Rice Krispies, 1box of Coco Pops, 1 box of Shreddies, and the remainder are Raisin Bran. Over a ten day period Brett plans to eat the contents of one box of cereal each morning. How many different order are possible if on the first day he has Raisin Bran?
3CF, 2 RK, 1 CP, 1 S, 3 RB (will only have 2 after first day)
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Consider the following problem:
Example 4: A city centre has a rectangular road system with streets running north to south and avenues running west to east.
a. below a grid represents the situation
Sean
b. Sean is driving a car and is situated at the extreme northwest corner of
the city centre. In how many ways can he drive to the extreme southeast corner if at each turn he moves closer to his destination (assume all streets and avenues allow two –way traffic)
Example 5: Find the number of pathways from A to B if paths must always move closer to B.
A
B
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A
B
Example 6: How many different routes are there from A to B if you only travel south or east.
Section 2.5:Exploring Combinations Combinations is a grouping of objects where order does not matter. This means for two objects “ab” would be the same as “ba” Example 1: given the letters a,b and c
A) How many permutations are possible using two of the three letters?
B) How many combinations are possible using two of the three letters?
Practice problems: Page 104-107 #1-7, 9-12, 15-17 (omit 3,10,11a)
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When is order not important?
1. From a group of four students, three are to be elected to an executive committee with a
specific position. The positions are as follows:
1st position President
2nd position Vice President
3rd position Treasurer
a. Does the order in which the students are elected matter? Why?
b. In how many ways can the positions be filled from this group?
2. Now suppose that the three students are to be selected to serve on a committee.
a. Is the order in which the three students are selected still important? Why or
why not?
b. How many possible committees from the group of four students are now
possible?
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3. You are part of a group of 6 students.
a. How many handshakes are possible if each student shakes every other
student’s hand once?
4. Part 1 deals with permutations, part 2 and 3 dealt with combinations.
What is the biggest difference between a permutation and a combination?
Example 1: Identify the following as a combination or permutation
A) The number of ways that three horses out of eight could end up first, second and third.
B) There are 16 students in a class, the number of ways that 4 students can be selected to serve on a committee.
C) The number of ways to pick a group of 2 boys and 3 girls from a class of 14 boys and 15 girls.
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Section 2.6:Combinations • An arrangement of a group of objects when order does not matter.
• There are several ways to find the number of possible combination.
1) use reasoning. Use the fundamental counting principle and divide by the number of ways that the object can be arranged among themselves.
For example, calculate the number of combinations of three digits made from the digits 1, 2, 3, 4, and 5:
5 x 4 x 3 = 60
However, 3 digits can be arranged 3! ways among themselves. So:
603! = 10
2) Use the Combination formula :
Calculate the number of combinations of three digits made from the digits 1, 2, 3, 4, and 5:
NOTE: The n rC key on the calculator can be used to evaluate combinations:
The number of combinations of “n” items take “r” at a time is:
( )!
! !n r
n nCr n r r
= = −
where 0 ≤ 𝑟 ≤ 𝑛
In some texts n rC is written as nr
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Example 1: Three students from a class of 10 are to be chosen to go on a school trip. In how many ways can they be selected? Example 2: To win LOTTO 6-49 a person must correctly choose six numbers from 1 to 49. Jasper, wanting to play LOTTO 6-49, began to wonder how many numbers he could make up. How many choices would Jasper have to make to ensure he had the six winning numbers?
Example 3: The Athletic Council decides to form a sub-committee of seven council members to look at how funds raised should be spent on sports activities in the school. There are a total of 15 athletic council members, 9 males and 6 females. The sub-committee must consist of exactly 3 females.
a. In how many ways can the females be chosen?
b. In how many ways can the males be chosen?
c. In how many ways can the sub-committee be chosen?
d. In how many ways can the sub-committee be chosen if Bruce the football coach must be included?
Practice problems: Page 118 #1,3,4,8
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Example 4: A group of people consists of 3 males and 4 females.
a) How many different committees of 3 people can be formed with 1 male and 2 females?
b) How many different committees of 5 people can be formed with exactly 3 females?
c) How many different committees of 3 people can be formed with at least one male on the committee?
Practice problems: Page 118-119 #8,10,11
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Combinations which are equivalent
Example5: Jane calculated 10 2C to be 45 arrangements. She then calculated 10 8C to be 45 arrangements. Use factorial notation to prove this:
10 2 10 8C C=
10!2!8!
10!8!2!
10 9 8!2!8!• • 10 9 8!
8!2!• •
45 = 45
Solving for “n” in Combinations Problems
Example 6: Solve nC2 = 10
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Example 7: During a Pee Wee hockey tryout, all the players met on the ice after the last practice and shook hands with each other. How many players attended the tryouts if there were 300 handshakes in all?
Solution:
A handshake requires 2 people nC2 = 300
Practice problems: Page 118-119 #9a,10,11,13a,15a
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2.6 Extra Practice Combinations:
1. A committee consists of 10 people. (A) How many ways can a subcommittee of 3 people be selected? (B) How many ways can an executive subcommittee of three people (chairman, treasurer and secretary) be selected? (C) Why are the answers to parts (A) and (B) different?
2. In the Powerball Lottery in Oregon, you must choose 5 numbers from 1 to 49 and 1 number from 1-42. How many different ways are there to choose the numbers?
3. There are 8 boys and 12 girls in a drama club. How many ways can a committee of 5 be selected if A) there are exactly 3 females?
B) there are at least 3 boys?
4. Set up the appropriate equation and use it to algebraically solve for n:
(A) 101 =Cn B) 212 =Cn C) 61 =−nn C
D) 1511 =−+ nn C E) 12012 =Cn
ANSWERS: 1.a)120 b)720 c)order matters in b. 2. 800889128 3. A)6160 B) 4592 4.A) 10 B) 7 C) 6 D) 5 D) 120
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Section 2.7:Solving Counting Problems Reminder: not every problem is a permutation or combination. Some problems
require the fundamental counting principle
When solving counting problems, you need to determine if order plays a role in the
situation. Once this is established, you can use the appropriate permutation or
combination formula.
Once you have established whether a problem involves permutations or
combinations you can also use these strategies:
• Look for conditions or restrictions. Consider these first as you develop your
solution
• If there is a repetition of r of the n objects to be eliminated, it is usually done
by dividing by r!
• If a problem involves multiple tasks that are connected by the word AND,
then the fundamental counting principle can be applied: multiply the number
of ways that each task can occur
• If a problem involves multiple tasks that are connected by the word OR (at
most, at least), the fundamental counting principle does not apply: add the
number of ways that each task can occur. This typically is found in counting
problems that involve several cases.
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Example 1: There are 20 athletes competing in a triathlon.
A) In how many ways can the top3 medal winners be chosen?
B) In how many ways can 3 athletes be chosen for an interview? Example 2: Six actors and eight actresses are available for a play with four male roles and three female roles. How many different cast lists are possible?
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Example 3: The guidance councilor and the leadership team are having a group photo taken. There are four boys and six girls. The photographer wants the boys to sit together and the girls to sit together for one of the poses. How many ways can the group sit in a row of 11 chairs for this pose?
Example 4: Find the number of different arrangements of the letters in the word
ANSWER under each condition:
a. that begin with an S
b. that begin with a vowel and end with a consonant
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Example 5: Consider a standard deck of 52 cards. How many different four card hands have:
a. at least 2 kings? (2K 2others) + (3K 1 other) + (4K 0 other)
b. at most 2 clubs? (0clubs 4 other) + (1 club 3 other) + (2 club 2 other)
Practice problems: Page 126-127 #1-3,5-7,9,11,13-14
